
theorem Th18:
  for X be non trivial ComplexNormSpace holds
  BoundedLinearOperatorsNorm(X,X).(id the carrier of X) = 1
proof
  let X be non trivial ComplexNormSpace;
  consider v be VECTOR of X such that
A1: ||.v.|| = 1 by Th17;
  reconsider ii=(id the carrier of X) as Lipschitzian LinearOperator of X,X
  by Th3;
A2: now
    let r be Real;
    assume r in PreNorms(ii);
    then ex t be VECTOR of X st r=||.ii.t.|| & ||.t.|| <= 1;
    hence r <=1;
  end;
  ii.v =v;
  then
A3: 1 in {||.ii.t.|| where t is VECTOR of X : ||.t.|| <= 1 } by A1;
  PreNorms(ii) is non empty bounded_above by CLOPBAN1:26;
  then
A4: 1 <=upper_bound PreNorms(ii) by A3,SEQ_4:def 1;
  (for s be Real st s in PreNorms(ii) holds s <= 1) implies upper_bound
  PreNorms(ii) <= 1 by SEQ_4:45;
  then upper_bound PreNorms(ii) =1 by A2,A4,XXREAL_0:1;
  hence thesis by CLOPBAN1:29;
end;
